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Theorem List for Metamath Proof Explorer - 16101-16200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-evls 16101* Define the evaluation map for the polynomial algebra. The function  ( (
I evalSub  S ) `  R
) : V --> ( S  ^m  ( S  ^m  I ) ) makes sense when  I is an index set,  S is a ring,  R is a subring of  S, and where  V is the set of polynomials in  ( I mPoly  R
). This function maps an element of the formal polynomial algebra (with coefficients in  R) to a function from assignments  I --> S of the variables to elements of  S formed by evaluating the polynomial with the given assignments. (Contributed by Stefan O'Rear, 11-Mar-2015.)
 |- evalSub  =  ( i  e.  _V ,  s  e.  CRing  |->  [_ ( Base `  s )  /  b ]_ ( r  e.  (SubRing `  s )  |-> 
 [_ ( i mPoly  (
 ss  r ) )  /  w ]_ ( iota_ f  e.  ( w RingHom  ( s  ^s  ( b  ^m  i ) ) ) ( ( f  o.  (algSc `  w ) )  =  ( x  e.  r  |->  ( ( b  ^m  i )  X.  { x } ) )  /\  ( f  o.  (
 i mVar  ( ss  r ) ) )  =  ( x  e.  i  |->  ( g  e.  ( b 
 ^m  i )  |->  ( g `  x ) ) ) ) ) ) )
 
Definitiondf-evl 16102* A simplication of evalSub when the evaluation ring is the same as the coefficient ring. (Contributed by Stefan O'Rear, 19-Mar-2015.)
 |- eval  =  ( i  e.  _V ,  r  e.  _V  |->  ( ( i evalSub  r
 ) `  ( Base `  r ) ) )
 
Definitiondf-mhp 16103* Define the subspaces of order-  n homogeneous polynomials. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |- mHomP  =  ( i  e.  _V ,  r  e.  _V  |->  ( n  e.  NN0  |->  { f  e.  ( Base `  ( i mPoly  r ) )  |  ( `' f " ( _V  \  { ( 0g `  r ) } )
 )  C_  { g  e.  { h  e.  ( NN0  ^m  i )  |  ( `' h " NN )  e.  Fin }  |  sum_ j  e.  NN0  ( g `  j
 )  =  n } } ) )
 
Definitiondf-psd 16104* Define the differentiation operation on multivariate polynomials. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |- mPSDer  =  ( i  e.  _V ,  r  e.  _V  |->  ( x  e.  i  |->  ( f  e.  ( Base `  ( i mPwSer  r
 ) )  |->  ( k  e.  { h  e.  ( NN0  ^m  i
 )  |  ( `' h " NN )  e.  Fin }  |->  ( ( ( k `  x )  +  1 )
 (.g `  r ) ( f `  ( k  o F  +  (
 y  e.  i  |->  if ( y  =  x ,  1 ,  0 ) ) ) ) ) ) ) ) )
 
Definitiondf-ltbag 16105* Define a well-order on the set of all finite bags from the index set  i given a wellordering  r of  i. (Contributed by Mario Carneiro, 8-Feb-2015.)
 |- 
 <bag 
 =  ( r  e. 
 _V ,  i  e. 
 _V  |->  { <. x ,  y >.  |  ( { x ,  y }  C_  { h  e.  ( NN0  ^m  i
 )  |  ( `' h " NN )  e.  Fin }  /\  E. z  e.  i  (
 ( x `  z
 )  <  ( y `  z )  /\  A. w  e.  i  (
 z r w  ->  ( x `  w )  =  ( y `  w ) ) ) ) } )
 
Definitiondf-opsr 16106* Define a total order on the set of all power series in  s from the index set  i given a wellordering  r of  i and a totally ordered base ring  s. (Contributed by Mario Carneiro, 8-Feb-2015.)
 |- ordPwSer  =  ( i  e.  _V ,  s  e.  _V  |->  ( r  e.  ~P ( i  X.  i
 )  |->  [_ ( i mPwSer  s
 )  /  p ]_ ( p sSet  <. ( le `  ndx ) ,  { <. x ,  y >.  |  ( { x ,  y }  C_  ( Base `  p )  /\  ( [. { h  e.  ( NN0  ^m  i
 )  |  ( `' h " NN )  e.  Fin }  /  d ]. E. z  e.  d  ( ( x `  z ) ( lt `  s ) ( y `
  z )  /\  A. w  e.  d  ( w ( r  <bag  i ) z  ->  ( x `  w )  =  ( y `  w ) ) )  \/  x  =  y ) ) } >. ) ) )
 
Definitiondf-selv 16107* Define the "variable selection" function. The function  ( (
I selectVars  R ) `  J
) maps elements of  ( I mPoly  R ) bijectively onto  ( J mPoly  ( ( I  \  J ) mPoly 
R ) ) in the natural way, for example if  I  =  { x ,  y } and  J  =  { y } it would map  1  +  x  +  y  +  x
y  e.  ( { x ,  y } mPoly 
ZZ ) to  ( 1  +  x )  +  ( 1  +  x ) y  e.  ( { y } mPoly  ( {
x } mPoly  ZZ )
). This, for example, allows one to treat a multivariate polynomial as a univariate polynomial with coefficients in a polynomial ring with one less variable. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |- selectVars  =  ( i  e.  _V ,  r  e.  _V  |->  ( j  e.  ~P i  |->  ( f  e.  ( i mPoly  r ) 
 |->  [_ ( ( i 
 \  j ) mPoly  r
 )  /  s ]_ [_ ( x  e.  (Scalar `  s )  |->  ( x ( .s `  s
 ) ( 1r `  s ) ) ) 
 /  c ]_ (
 ( ( ( i evalSub  s ) `  (
 c  "s  r ) ) `  ( c  o.  f
 ) ) `  ( x  e.  i  |->  if ( x  e.  j ,  ( ( j mVar  (
 ( i  \  j
 ) mPoly  r ) ) `  x ) ,  (
 c  o.  ( ( ( i  \  j
 ) mVar  r ) `  x ) ) ) ) ) ) ) )
 
Definitiondf-algind 16108* Define the predicate "the set  v is algebraically independent in the algebra  w". A collection of vectors is algebraically independent if no nontrivial polynomial with elements from the subset evaluates to zero. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |- AlgInd  =  ( w  e.  _V ,  k  e.  ~P ( Base `  w )  |->  { v  e.  ~P ( Base `  w )  |  Fun  `' ( f  e.  ( Base `  (
 v mPoly  ( ws  k ) ) ) 
 |->  ( ( ( ( v evalSub  w ) `  k
 ) `  f ) `  (  _I  |`  v ) ) ) } )
 
Theoremreldmpsr 16109 The multivariate power series constructor is a proper binary operator. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |- 
 Rel  dom mPwSer
 
Theorempsrval 16110* Value of the multivariate power series structure. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  K  =  (
 Base `  R )   &    |-  .+  =  ( +g  `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  O  =  ( TopOpen `  R )   &    |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }   &    |-  ( ph  ->  B  =  ( K  ^m  D ) )   &    |-  .+b  =  (  o F  .+  |`  ( B  X.  B ) )   &    |-  .X. 
 =  ( f  e.  B ,  g  e.  B  |->  ( k  e.  D  |->  ( R  gsumg  ( x  e.  { y  e.  D  |  y  o R  <_  k }  |->  ( ( f `  x )  .x.  ( g `
  ( k  o F  -  x ) ) ) ) ) ) )   &    |-  .xb  =  ( x  e.  K ,  f  e.  B  |->  ( ( D  X.  { x } )  o F  .x.  f ) )   &    |-  ( ph  ->  J  =  (
 Xt_ `  ( D  X.  { O } )
 ) )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  e.  X )   =>    |-  ( ph  ->  S  =  ( { <. ( Base ` 
 ndx ) ,  B >. ,  <. ( +g  `  ndx ) ,  .+b  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s
 `  ndx ) ,  .xb  >. ,  <. (TopSet `  ndx ) ,  J >. } ) )
 
Theorempsrvalstr 16111 The multivariate power series structure is a function. (Contributed by Mario Carneiro, 8-Feb-2015.)
 |-  ( { <. ( Base ` 
 ndx ) ,  B >. ,  <. ( +g  `  ndx ) ,  .+  >. ,  <. ( .r `  ndx ) ,  .X.  >. }  u.  { <. (Scalar `  ndx ) ,  R >. ,  <. ( .s
 `  ndx ) ,  .x.  >. ,  <. (TopSet `  ndx ) ,  J >. } ) Struct  <. 1 ,  9 >.
 
Theorempsrbag 16112* Elementhood in the set of finite bags. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   =>    |-  ( I  e.  V  ->  ( F  e.  D  <->  ( F : I --> NN0  /\  ( `' F " NN )  e.  Fin ) ) )
 
Theorempsrbagf 16113* A finite bag is a function. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   =>    |-  ( ( I  e.  V  /\  F  e.  D )  ->  F : I
 --> NN0 )
 
Theorempsrbaglesupp 16114* The support of a dominated bag is smaller than the dominating bag. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   =>    |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F ) ) 
 ->  ( `' G " NN )  C_  ( `' F " NN )
 )
 
Theorempsrbaglecl 16115* The set of finite bags is downward-closed. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   =>    |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F ) ) 
 ->  G  e.  D )
 
Theorempsrbagaddcl 16116* The sum of two finite bags is a finite bag. (Contributed by Mario Carneiro, 9-Jan-2015.)
 |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   =>    |-  ( ( I  e.  V  /\  F  e.  D  /\  G  e.  D )  ->  ( F  o F  +  G )  e.  D )
 
Theorempsrbagcon 16117* The analogue of the statement " 0  <_  G  <_  F implies  0  <_  F  -  G  <_  F " for finite bags. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   =>    |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  o R  <_  F ) ) 
 ->  ( ( F  o F  -  G )  e.  D  /\  ( F  o F  -  G )  o R  <_  F ) )
 
Theorempsrbaglefi 16118* There are finitely many bags dominated by a given bag. (Contributed by Mario Carneiro, 29-Dec-2014.) (Revised by Mario Carneiro, 25-Jan-2015.)
 |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   =>    |-  ( ( I  e.  V  /\  F  e.  D )  ->  { y  e.  D  |  y  o R  <_  F }  e.  Fin )
 
Theorempsrbagconcl 16119* The complement of a bag is a bag. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  S  =  {
 y  e.  D  |  y  o R  <_  F }   =>    |-  ( ( I  e.  V  /\  F  e.  D  /\  X  e.  S )  ->  ( F  o F  -  X )  e.  S )
 
Theorempsrbagconf1o 16120* Bag complementation is a bijection on the set of bags dominated by a given bag  F. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  S  =  {
 y  e.  D  |  y  o R  <_  F }   =>    |-  ( ( I  e.  V  /\  F  e.  D )  ->  ( x  e.  S  |->  ( F  o F  -  x ) ) : S -1-1-onto-> S )
 
Theoremgsumbagdiaglem 16121* Lemma for gsumbagdiag 16122. (Contributed by Mario Carneiro, 5-Jan-2015.)
 |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  S  =  {
 y  e.  D  |  y  o R  <_  F }   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  F  e.  D )   =>    |-  ( ( ph  /\  ( X  e.  S  /\  Y  e.  { x  e.  D  |  x  o R  <_  ( F  o F  -  X ) }
 ) )  ->  ( Y  e.  S  /\  X  e.  { x  e.  D  |  x  o R  <_  ( F  o F  -  Y ) }
 ) )
 
Theoremgsumbagdiag 16122* Two-dimensional commutation of a group sum over a "triangular" region. fsum0diag 12240 analogue for finite bags. (Contributed by Mario Carneiro, 5-Jan-2015.)
 |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  S  =  {
 y  e.  D  |  y  o R  <_  F }   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  F  e.  D )   &    |-  B  =  ( Base `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  (
 ( ph  /\  ( j  e.  S  /\  k  e.  { x  e.  D  |  x  o R  <_  ( F  o F  -  j ) } )
 )  ->  X  e.  B )   =>    |-  ( ph  ->  ( G  gsumg  ( j  e.  S ,  k  e.  { x  e.  D  |  x  o R  <_  ( F  o F  -  j ) }  |->  X ) )  =  ( G  gsumg  ( k  e.  S ,  j  e.  { x  e.  D  |  x  o R  <_  ( F  o F  -  k ) }  |->  X ) ) )
 
Theorempsrass1lem 16123* A group sum commutation used by psrass1 16150. (Contributed by Mario Carneiro, 5-Jan-2015.)
 |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  S  =  {
 y  e.  D  |  y  o R  <_  F }   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  F  e.  D )   &    |-  B  =  ( Base `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  (
 ( ph  /\  ( j  e.  S  /\  k  e.  { x  e.  D  |  x  o R  <_  ( F  o F  -  j ) } )
 )  ->  X  e.  B )   &    |-  ( k  =  ( n  o F  -  j )  ->  X  =  Y )   =>    |-  ( ph  ->  ( G  gsumg  ( n  e.  S  |->  ( G  gsumg  ( j  e.  { x  e.  D  |  x  o R  <_  n }  |->  Y ) ) ) )  =  ( G  gsumg  ( j  e.  S  |->  ( G  gsumg  ( k  e.  { x  e.  D  |  x  o R  <_  ( F  o F  -  j
 ) }  |->  X ) ) ) ) )
 
Theorempsrbas 16124* The base set of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  K  =  (
 Base `  R )   &    |-  D  =  { f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }   &    |-  B  =  ( Base `  S )   &    |-  ( ph  ->  I  e.  V )   =>    |-  ( ph  ->  B  =  ( K  ^m  D ) )
 
Theorempsrelbas 16125* An element of the set of power series is a function on the coefficients. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  K  =  (
 Base `  R )   &    |-  D  =  { f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }   &    |-  B  =  ( Base `  S )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  X : D --> K )
 
Theorempsrplusg 16126 The addition operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  B  =  (
 Base `  S )   &    |-  .+  =  ( +g  `  R )   &    |-  .+b  =  ( +g  `  S )   =>    |-  .+b  =  (  o F  .+  |`  ( B  X.  B ) )
 
Theorempsradd 16127 The addition operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  B  =  (
 Base `  S )   &    |-  .+  =  ( +g  `  R )   &    |-  .+b  =  ( +g  `  S )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X  .+b  Y )  =  ( X  o F  .+  Y ) )
 
Theorempsraddcl 16128 Closure of the power series addition operation. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  B  =  (
 Base `  S )   &    |-  .+  =  ( +g  `  S )   &    |-  ( ph  ->  R  e.  Grp )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X  .+  Y )  e.  B )
 
Theorempsrmulr 16129* The multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  B  =  (
 Base `  S )   &    |-  .x.  =  ( .r `  R )   &    |-  .xb 
 =  ( .r `  S )   &    |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }   =>    |-  .xb  =  ( f  e.  B ,  g  e.  B  |->  ( k  e.  D  |->  ( R  gsumg  ( x  e.  { y  e.  D  |  y  o R  <_  k }  |->  ( ( f `  x )  .x.  ( g `
  ( k  o F  -  x ) ) ) ) ) ) )
 
Theorempsrmulfval 16130* The multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  B  =  (
 Base `  S )   &    |-  .x.  =  ( .r `  R )   &    |-  .xb 
 =  ( .r `  S )   &    |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  G  e.  B )   =>    |-  ( ph  ->  ( F  .xb  G )  =  ( k  e.  D  |->  ( R  gsumg  ( x  e.  {
 y  e.  D  |  y  o R  <_  k }  |->  ( ( F `
  x )  .x.  ( G `  ( k  o F  -  x ) ) ) ) ) ) )
 
Theorempsrmulval 16131* The multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  B  =  (
 Base `  S )   &    |-  .x.  =  ( .r `  R )   &    |-  .xb 
 =  ( .r `  S )   &    |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  G  e.  B )   &    |-  ( ph  ->  X  e.  D )   =>    |-  ( ph  ->  (
 ( F  .xb  G ) `
  X )  =  ( R  gsumg  ( k  e.  {
 y  e.  D  |  y  o R  <_  X }  |->  ( ( F `
  k )  .x.  ( G `  ( X  o F  -  k
 ) ) ) ) ) )
 
Theorempsrmulcllem 16132* Closure of the power series multiplication operation. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  B  =  (
 Base `  S )   &    |-  .x.  =  ( .r `  S )   &    |-  ( ph  ->  R  e.  Ring
 )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  D  =  { f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }   =>    |-  ( ph  ->  ( X  .x.  Y )  e.  B )
 
Theorempsrmulcl 16133 Closure of the power series multiplication operation. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  B  =  (
 Base `  S )   &    |-  .x.  =  ( .r `  S )   &    |-  ( ph  ->  R  e.  Ring
 )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X  .x.  Y )  e.  B )
 
Theorempsrsca 16134 The scalar field of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  W )   =>    |-  ( ph  ->  R  =  (Scalar `  S )
 )
 
Theorempsrvscafval 16135* The scalar multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  .xb  =  ( .s `  S )   &    |-  K  =  ( Base `  R )   &    |-  B  =  ( Base `  S )   &    |-  .x.  =  ( .r `  R )   &    |-  D  =  { h  e.  ( NN0  ^m  I
 )  |  ( `' h " NN )  e.  Fin }   =>    |-  .xb  =  ( x  e.  K ,  f  e.  B  |->  ( ( D  X.  { x }
 )  o F  .x.  f ) )
 
Theorempsrvsca 16136* The scalar multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  .xb  =  ( .s `  S )   &    |-  K  =  ( Base `  R )   &    |-  B  =  ( Base `  S )   &    |-  .x.  =  ( .r `  R )   &    |-  D  =  { h  e.  ( NN0  ^m  I
 )  |  ( `' h " NN )  e.  Fin }   &    |-  ( ph  ->  X  e.  K )   &    |-  ( ph  ->  F  e.  B )   =>    |-  ( ph  ->  ( X  .xb  F )  =  ( ( D  X.  { X } )  o F  .x.  F )
 )
 
Theorempsrvscaval 16137* The scalar multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  .xb  =  ( .s `  S )   &    |-  K  =  ( Base `  R )   &    |-  B  =  ( Base `  S )   &    |-  .x.  =  ( .r `  R )   &    |-  D  =  { h  e.  ( NN0  ^m  I
 )  |  ( `' h " NN )  e.  Fin }   &    |-  ( ph  ->  X  e.  K )   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  Y  e.  D )   =>    |-  ( ph  ->  (
 ( X  .xb  F ) `
  Y )  =  ( X  .x.  ( F `  Y ) ) )
 
Theorempsrvscacl 16138 Closure of the power series scalar multiplication operation. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  .x.  =  ( .s `  S )   &    |-  K  =  ( Base `  R )   &    |-  B  =  ( Base `  S )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  X  e.  K )   &    |-  ( ph  ->  F  e.  B )   =>    |-  ( ph  ->  ( X  .x.  F )  e.  B )
 
Theorempsr0cl 16139* The zero element of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Grp )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  .0.  =  ( 0g `  R )   &    |-  B  =  ( Base `  S )   =>    |-  ( ph  ->  ( D  X.  {  .0.  }
 )  e.  B )
 
Theorempsr0lid 16140* The zero element of the ring of power series is a left identity. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Grp )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  .0.  =  ( 0g `  R )   &    |-  B  =  ( Base `  S )   &    |-  .+  =  ( +g  `  S )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  (
 ( D  X.  {  .0.  } )  .+  X )  =  X )
 
Theorempsrnegcl 16141* The negative function in the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Grp )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  N  =  ( inv g `  R )   &    |-  B  =  ( Base `  S )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  ( N  o.  X )  e.  B )
 
Theorempsrlinv 16142* The negative function in the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Grp )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  N  =  ( inv g `  R )   &    |-  B  =  ( Base `  S )   &    |-  ( ph  ->  X  e.  B )   &    |-  .0.  =  ( 0g `  R )   &    |- 
 .+  =  ( +g  `  S )   =>    |-  ( ph  ->  (
 ( N  o.  X )  .+  X )  =  ( D  X.  {  .0.  } ) )
 
Theorempsrgrp 16143 The ring of power series is a group. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Grp )   =>    |-  ( ph  ->  S  e.  Grp )
 
Theorempsr0 16144* The zero element of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Grp )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  O  =  ( 0g `  R )   &    |-  .0.  =  ( 0g `  S )   =>    |-  ( ph  ->  .0.  =  ( D  X.  { O } ) )
 
Theorempsrneg 16145* The negative function of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Grp )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  N  =  ( inv g `  R )   &    |-  B  =  ( Base `  S )   &    |-  M  =  ( inv g `  S )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  ( M `  X )  =  ( N  o.  X ) )
 
Theorempsrlmod 16146 The ring of power series is a left module. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Ring )   =>    |-  ( ph  ->  S  e.  LMod
 )
 
Theorempsr1cl 16147* The identity element of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  .0.  =  ( 0g `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  U  =  ( x  e.  D  |->  if ( x  =  ( I  X.  { 0 } ) ,  .1.  ,  .0.  ) )   &    |-  B  =  ( Base `  S )   =>    |-  ( ph  ->  U  e.  B )
 
Theorempsrlidm 16148* The identity element of the ring of power series is a left identity. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  .0.  =  ( 0g `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  U  =  ( x  e.  D  |->  if ( x  =  ( I  X.  { 0 } ) ,  .1.  ,  .0.  ) )   &    |-  B  =  ( Base `  S )   &    |-  .x.  =  ( .r `  S )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  ( U  .x.  X )  =  X )
 
Theorempsrridm 16149* The identity element of the ring of power series is a right identity. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  .0.  =  ( 0g `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  U  =  ( x  e.  D  |->  if ( x  =  ( I  X.  { 0 } ) ,  .1.  ,  .0.  ) )   &    |-  B  =  ( Base `  S )   &    |-  .x.  =  ( .r `  S )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  ( X  .x.  U )  =  X )
 
Theorempsrass1 16150* Associative identity for the ring of power series. (Contributed by Mario Carneiro, 5-Jan-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  .X.  =  ( .r `  S )   &    |-  B  =  ( Base `  S )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   =>    |-  ( ph  ->  ( ( X  .X.  Y )  .X.  Z )  =  ( X  .X.  ( Y  .X.  Z ) ) )
 
Theorempsrdi 16151* Distributive law for the ring of power series. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  .X.  =  ( .r `  S )   &    |-  B  =  ( Base `  S )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   &    |-  .+  =  ( +g  `  S )   =>    |-  ( ph  ->  ( X  .X.  ( Y  .+  Z ) )  =  ( ( X  .X.  Y )  .+  ( X  .X.  Z ) ) )
 
Theorempsrdir 16152* Distributive law for the ring of power series. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  .X.  =  ( .r `  S )   &    |-  B  =  ( Base `  S )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   &    |-  .+  =  ( +g  `  S )   =>    |-  ( ph  ->  ( ( X 
 .+  Y )  .X.  Z )  =  ( ( X  .X.  Z )  .+  ( Y  .X.  Z ) ) )
 
Theorempsrcom 16153* Commutative law for the ring of power series. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  .X.  =  ( .r `  S )   &    |-  B  =  ( Base `  S )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  R  e.  CRing )   =>    |-  ( ph  ->  ( X  .X.  Y )  =  ( Y  .X.  X ) )
 
Theorempsrass23 16154* Associative identities for the ring of power series. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  .X.  =  ( .r `  S )   &    |-  B  =  ( Base `  S )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  R  e.  CRing )   &    |-  K  =  (
 Base `  R )   &    |-  .x.  =  ( .s `  S )   &    |-  ( ph  ->  A  e.  K )   =>    |-  ( ph  ->  (
 ( ( A  .x.  X )  .X.  Y )  =  ( A  .x.  ( X  .X.  Y ) ) 
 /\  ( X  .X.  ( A  .x.  Y ) )  =  ( A 
 .x.  ( X  .X.  Y ) ) ) )
 
Theorempsrrng 16155 The ring of power series is a ring. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Ring )   =>    |-  ( ph  ->  S  e.  Ring
 )
 
Theorempsr1 16156* The identity element of the ring of power series. (Contributed by Mario Carneiro, 8-Jan-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  .0.  =  ( 0g `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  U  =  ( 1r `  S )   =>    |-  ( ph  ->  U  =  ( x  e.  D  |->  if ( x  =  ( I  X.  { 0 } ) ,  .1.  ,  .0.  ) ) )
 
Theorempsrcrng 16157 The ring of power series is commutative ring. (Contributed by Mario Carneiro, 10-Jan-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  CRing )   =>    |-  ( ph  ->  S  e.  CRing
 )
 
Theorempsrassa 16158 The ring of power series is an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  CRing )   =>    |-  ( ph  ->  S  e. AssAlg )
 
Theoremresspsrbas 16159 A restricted power series algebra has the same base set. (Contributed by Mario Carneiro, 3-Jul-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  H  =  ( Rs  T )   &    |-  U  =  ( I mPwSer  H )   &    |-  B  =  (
 Base `  U )   &    |-  P  =  ( Ss  B )   &    |-  ( ph  ->  T  e.  (SubRing `  R ) )   =>    |-  ( ph  ->  B  =  ( Base `  P )
 )
 
Theoremresspsradd 16160 A restricted power series algebra has the same addition operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  H  =  ( Rs  T )   &    |-  U  =  ( I mPwSer  H )   &    |-  B  =  (
 Base `  U )   &    |-  P  =  ( Ss  B )   &    |-  ( ph  ->  T  e.  (SubRing `  R ) )   =>    |-  ( ( ph  /\  ( X  e.  B  /\  Y  e.  B )
 )  ->  ( X ( +g  `  U ) Y )  =  ( X ( +g  `  P ) Y ) )
 
Theoremresspsrmul 16161 A restricted power series algebra has the same multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  H  =  ( Rs  T )   &    |-  U  =  ( I mPwSer  H )   &    |-  B  =  (
 Base `  U )   &    |-  P  =  ( Ss  B )   &    |-  ( ph  ->  T  e.  (SubRing `  R ) )   =>    |-  ( ( ph  /\  ( X  e.  B  /\  Y  e.  B )
 )  ->  ( X ( .r `  U ) Y )  =  ( X ( .r `  P ) Y ) )
 
Theoremresspsrvsca 16162 A restricted power series algebra has the same scalar multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  H  =  ( Rs  T )   &    |-  U  =  ( I mPwSer  H )   &    |-  B  =  (
 Base `  U )   &    |-  P  =  ( Ss  B )   &    |-  ( ph  ->  T  e.  (SubRing `  R ) )   =>    |-  ( ( ph  /\  ( X  e.  T  /\  Y  e.  B )
 )  ->  ( X ( .s `  U ) Y )  =  ( X ( .s `  P ) Y ) )
 
Theoremsubrgpsr 16163 A subring of the base ring induces a subring of power series. (Contributed by Mario Carneiro, 3-Jul-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  H  =  ( Rs  T )   &    |-  U  =  ( I mPwSer  H )   &    |-  B  =  (
 Base `  U )   =>    |-  ( ( I  e.  V  /\  T  e.  (SubRing `  R )
 )  ->  B  e.  (SubRing `  S ) )
 
Theoremmvridlem 16164* A bag containing one element is a finite bag. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  D  =  { h  e.  ( NN0  ^m  I
 )  |  ( `' h " NN )  e.  Fin }   =>    |-  ( I  e.  V  ->  ( y  e.  I  |->  if ( y  =  X ,  1 ,  0 ) )  e.  D )
 
Theoremmvrfval 16165* Value of the generating elements of the power series structure. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  V  =  ( I mVar 
 R )   &    |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }   &    |-  .0.  =  ( 0g `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  e.  Y )   =>    |-  ( ph  ->  V  =  ( x  e.  I  |->  ( f  e.  D  |->  if ( f  =  ( y  e.  I  |->  if ( y  =  x ,  1 ,  0 ) ) ,  .1.  ,  .0.  ) ) ) )
 
Theoremmvrval 16166* Value of the generating elements of the power series structure. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  V  =  ( I mVar 
 R )   &    |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }   &    |-  .0.  =  ( 0g `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  e.  Y )   &    |-  ( ph  ->  X  e.  I )   =>    |-  ( ph  ->  ( V `  X )  =  ( f  e.  D  |->  if ( f  =  ( y  e.  I  |->  if ( y  =  X ,  1 ,  0 ) ) ,  .1.  ,  .0.  ) ) )
 
Theoremmvrval2 16167* Value of the generating elements of the power series structure. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  V  =  ( I mVar 
 R )   &    |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }   &    |-  .0.  =  ( 0g `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  e.  Y )   &    |-  ( ph  ->  X  e.  I )   &    |-  ( ph  ->  F  e.  D )   =>    |-  ( ph  ->  ( ( V `  X ) `  F )  =  if ( F  =  ( y  e.  I  |->  if ( y  =  X ,  1 ,  0 ) ) ,  .1.  ,  .0.  )
 )
 
Theoremmvrid 16168* The  X i-th coefficient of the term  X i is  1. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  V  =  ( I mVar 
 R )   &    |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }   &    |-  .0.  =  ( 0g `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  e.  Y )   &    |-  ( ph  ->  X  e.  I )   =>    |-  ( ph  ->  (
 ( V `  X ) `  ( y  e.  I  |->  if ( y  =  X ,  1 ,  0 ) ) )  =  .1.  )
 
Theoremmvrf 16169 The power series variable function is a function from the index set to elements of the power series structure representing  X
i for each  i. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  V  =  ( I mVar  R )   &    |-  B  =  ( Base `  S )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  e.  Ring )   =>    |-  ( ph  ->  V : I --> B )
 
Theoremmvrf1 16170 The power series variable function is injective if the base ring is nonzero. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  V  =  ( I mVar  R )   &    |-  B  =  ( Base `  S )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  .0.  =  ( 0g `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  ( ph  ->  .1. 
 =/=  .0.  )   =>    |-  ( ph  ->  V : I -1-1-> B )
 
Theoremmvrcl2 16171 A power series variable is an element of the base set. (Contributed by Mario Carneiro, 29-Dec-2014.)
 |-  S  =  ( I mPwSer  R )   &    |-  V  =  ( I mVar  R )   &    |-  B  =  ( Base `  S )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  X  e.  I )   =>    |-  ( ph  ->  ( V `  X )  e.  B )
 
Theoremreldmmpl 16172 The multivariate polynomial constructor is a proper binary operator. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |- 
 Rel  dom mPoly
 
Theoremmplval 16173* Value of the set of multivariate polynomials. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  S  =  ( I mPwSer  R )   &    |-  B  =  (
 Base `  S )   &    |-  .0.  =  ( 0g `  R )   &    |-  U  =  { f  e.  B  |  ( `' f " ( _V  \  {  .0.  } )
 )  e.  Fin }   =>    |-  P  =  ( Ss  U )
 
Theoremmplbas 16174* Base set of the set of multivariate polynomials. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  S  =  ( I mPwSer  R )   &    |-  B  =  (
 Base `  S )   &    |-  .0.  =  ( 0g `  R )   &    |-  U  =  ( Base `  P )   =>    |-  U  =  { f  e.  B  |  ( `' f " ( _V  \  {  .0.  } )
 )  e.  Fin }
 
Theoremmplelbas 16175 Property of being a polynomial. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  S  =  ( I mPwSer  R )   &    |-  B  =  (
 Base `  S )   &    |-  .0.  =  ( 0g `  R )   &    |-  U  =  ( Base `  P )   =>    |-  ( X  e.  U  <->  ( X  e.  B  /\  ( `' X " ( _V  \  {  .0.  } )
 )  e.  Fin )
 )
 
Theoremmplval2 16176 Self-referential expression for the set of multivariate polynomials. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  S  =  ( I mPwSer  R )   &    |-  U  =  (
 Base `  P )   =>    |-  P  =  ( Ss  U )
 
Theoremmplbasss 16177 The set of polynomials is a subset of the set of power series. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  S  =  ( I mPwSer  R )   &    |-  U  =  (
 Base `  P )   &    |-  B  =  ( Base `  S )   =>    |-  U  C_  B
 
Theoremmplelf 16178* An polynomial is defined as a function on the coefficients. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  K  =  (
 Base `  R )   &    |-  B  =  ( Base `  P )   &    |-  D  =  { f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  X : D --> K )
 
Theoremmplsubglem 16179* If  A is an ideal of sets (a nonempty collection closed under subset and binary union) of the set  D of finite bags (the primary applications being  A  =  Fin and  A  =  ~P B for some  B), then the set of all power series whose coefficient functions are supported on an element of  A is a subgroup of the set of all power series. (Contributed by Mario Carneiro, 12-Jan-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  B  =  (
 Base `  S )   &    |-  .0.  =  ( 0g `  R )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  (/)  e.  A )   &    |-  ( ( ph  /\  ( x  e.  A  /\  y  e.  A )
 )  ->  ( x  u.  y )  e.  A )   &    |-  ( ( ph  /\  ( x  e.  A  /\  y  C_  x ) ) 
 ->  y  e.  A )   &    |-  ( ph  ->  U  =  { g  e.  B  |  ( `' g "
 ( _V  \  {  .0.  } ) )  e.  A } )   &    |-  ( ph  ->  R  e.  Grp )   =>    |-  ( ph  ->  U  e.  (SubGrp `  S )
 )
 
Theoremmpllsslem 16180* If  A is an ideal of subsets (a nonempty collection closed under subset and binary union) of the set  D of finite bags (the primary applications being  A  =  Fin and  A  =  ~P B for some  B), then the set of all power series whose coefficient functions are supported on an element of  A is a linear subspace of the set of all power series. (Contributed by Mario Carneiro, 12-Jan-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  B  =  (
 Base `  S )   &    |-  .0.  =  ( 0g `  R )   &    |-  D  =  { f  e.  ( NN0  ^m  I
 )  |  ( `' f " NN )  e.  Fin }   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  (/)  e.  A )   &    |-  ( ( ph  /\  ( x  e.  A  /\  y  e.  A )
 )  ->  ( x  u.  y )  e.  A )   &    |-  ( ( ph  /\  ( x  e.  A  /\  y  C_  x ) ) 
 ->  y  e.  A )   &    |-  ( ph  ->  U  =  { g  e.  B  |  ( `' g "
 ( _V  \  {  .0.  } ) )  e.  A } )   &    |-  ( ph  ->  R  e.  Ring )   =>    |-  ( ph  ->  U  e.  ( LSubSp `  S )
 )
 
Theoremmplsubg 16181 The set of polynomials is closed under addition, i.e. it is a subgroup of the set of power series. (Contributed by Mario Carneiro, 8-Jan-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  P  =  ( I mPoly  R )   &    |-  U  =  ( Base `  P )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  e.  Grp )   =>    |-  ( ph  ->  U  e.  (SubGrp `  S )
 )
 
Theoremmpllss 16182 The set of polynomials is closed under scalar multiplication, i.e. it is a linear subspace of the set of power series. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  P  =  ( I mPoly  R )   &    |-  U  =  ( Base `  P )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  e.  Ring )   =>    |-  ( ph  ->  U  e.  ( LSubSp `  S )
 )
 
Theoremmplsubrglem 16183* Lemma for mplsubrg 16184. (Contributed by Mario Carneiro, 9-Jan-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  P  =  ( I mPoly  R )   &    |-  U  =  ( Base `  P )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  D  =  {
 f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }   &    |-  .0.  =  ( 0g `  R )   &    |-  A  =  (  o F  +  " ( ( `' X " ( _V  \  {  .0.  } )
 )  X.  ( `' Y " ( _V  \  {  .0.  } ) ) ) )   &    |-  .x.  =  ( .r `  R )   &    |-  ( ph  ->  X  e.  U )   &    |-  ( ph  ->  Y  e.  U )   =>    |-  ( ph  ->  ( X ( .r `  S ) Y )  e.  U )
 
Theoremmplsubrg 16184 The set of polynomials is closed under multiplication, i.e. it is a subring of the set of power series. (Contributed by Mario Carneiro, 9-Jan-2015.)
 |-  S  =  ( I mPwSer  R )   &    |-  P  =  ( I mPoly  R )   &    |-  U  =  ( Base `  P )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  e.  Ring )   =>    |-  ( ph  ->  U  e.  (SubRing `  S )
 )
 
Theoremmpl0 16185* The zero polynomial. (Contributed by Mario Carneiro, 9-Jan-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  D  =  {
 f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }   &    |-  O  =  ( 0g `  R )   &    |-  .0.  =  ( 0g `  P )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  e.  Grp )   =>    |-  ( ph  ->  .0.  =  ( D  X.  { O } ) )
 
Theoremmpladd 16186 The addition operation on multivariate polynomials. (Contributed by Mario Carneiro, 9-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  B  =  (
 Base `  P )   &    |-  .+  =  ( +g  `  R )   &    |-  .+b  =  ( +g  `  P )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X  .+b  Y )  =  ( X  o F  .+  Y ) )
 
Theoremmplmul 16187* The multiplication operation on multivariate polynomials. (Contributed by Mario Carneiro, 9-Jan-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  B  =  (
 Base `  P )   &    |-  .x.  =  ( .r `  R )   &    |-  .xb 
 =  ( .r `  P )   &    |-  D  =  { h  e.  ( NN0  ^m  I )  |  ( `' h " NN )  e.  Fin }   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  G  e.  B )   =>    |-  ( ph  ->  ( F  .xb  G )  =  ( k  e.  D  |->  ( R  gsumg  ( x  e.  {
 y  e.  D  |  y  o R  <_  k }  |->  ( ( F `
  x )  .x.  ( G `  ( k  o F  -  x ) ) ) ) ) ) )
 
Theoremmpl1 16188* The identity element of the ring of polynomials. (Contributed by Mario Carneiro, 10-Jan-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  D  =  {
 f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }   &    |-  .0.  =  ( 0g `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  U  =  ( 1r `  P )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  e.  Ring )   =>    |-  ( ph  ->  U  =  ( x  e.  D  |->  if ( x  =  ( I  X.  { 0 } ) ,  .1.  ,  .0.  ) ) )
 
Theoremmplsca 16189 The scalar field of a multivariate polynomial structure. (Contributed by Mario Carneiro, 9-Jan-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  R  e.  W )   =>    |-  ( ph  ->  R  =  (Scalar `  P )
 )
 
Theoremmplvsca2 16190 The scalar multiplication operation on multivariate polynomials. (Contributed by Mario Carneiro, 9-Jan-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  S  =  ( I mPwSer  R )   &    |-  .x.  =  ( .s `  P )   =>    |-  .x.  =  ( .s `  S )
 
Theoremmplvsca 16191* The scalar multiplication operation on multivariate polynomials. (Contributed by Mario Carneiro, 9-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  .xb  =  ( .s `  P )   &    |-  K  =  ( Base `  R )   &    |-  B  =  ( Base `  P )   &    |-  .x.  =  ( .r `  R )   &    |-  D  =  { h  e.  ( NN0  ^m  I
 )  |  ( `' h " NN )  e.  Fin }   &    |-  ( ph  ->  X  e.  K )   &    |-  ( ph  ->  F  e.  B )   =>    |-  ( ph  ->  ( X  .xb  F )  =  ( ( D  X.  { X } )  o F  .x.  F )
 )
 
Theoremmplvscaval 16192* The scalar multiplication operation on multivariate polynomials. (Contributed by Mario Carneiro, 9-Jan-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  .xb  =  ( .s `  P )   &    |-  K  =  ( Base `  R )   &    |-  B  =  ( Base `  P )   &    |-  .x.  =  ( .r `  R )   &    |-  D  =  { h  e.  ( NN0  ^m  I
 )  |  ( `' h " NN )  e.  Fin }   &    |-  ( ph  ->  X  e.  K )   &    |-  ( ph  ->  F  e.  B )   &    |-  ( ph  ->  Y  e.  D )   =>    |-  ( ph  ->  (
 ( X  .xb  F ) `
  Y )  =  ( X  .x.  ( F `  Y ) ) )
 
Theoremmvrcl 16193 A power series variable is a polynomial. (Contributed by Mario Carneiro, 9-Jan-2015.)
 |-  P  =  ( I mPoly  R )   &    |-  V  =  ( I mVar  R )   &    |-  B  =  ( Base `  P )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  X  e.  I )   =>    |-  ( ph  ->  ( V `  X )  e.  B )
 
Theoremmplgrp 16194 The polynomial ring is a group. (Contributed by Mario Carneiro, 9-Jan-2015.)
 |-  P  =  ( I mPoly  R )   =>    |-  ( ( I  e.  V  /\  R  e.  Grp )  ->  P  e.  Grp )
 
Theoremmpllmod 16195 The polynomial ring is a left module. (Contributed by Mario Carneiro, 9-Jan-2015.)
 |-  P  =  ( I mPoly  R )   =>    |-  ( ( I  e.  V  /\  R  e.  Ring
 )  ->  P  e.  LMod
 )
 
Theoremmplrng 16196 The polynomial ring is a ring. (Contributed by Mario Carneiro, 9-Jan-2015.)
 |-  P  =  ( I mPoly  R )   =>    |-  ( ( I  e.  V  /\  R  e.  Ring
 )  ->  P  e.  Ring
 )
 
Theoremmplcrng 16197 The polynomial ring is a commutative ring. (Contributed by Mario Carneiro, 9-Jan-2015.)
 |-  P  =  ( I mPoly  R )   =>    |-  ( ( I  e.  V  /\  R  e.  CRing
 )  ->  P  e.  CRing
 )
 
Theoremmplassa 16198 The polynomial ring is an associative algebra. (Contributed by Mario Carneiro, 9-Jan-2015.)
 |-  P  =  ( I mPoly  R )   =>    |-  ( ( I  e.  V  /\  R  e.  CRing
 )  ->  P  e. AssAlg )
 
Theoremressmplbas2 16199 The base set of a restricted polynomial algebra consists of power series in the subring which are also polynomials (in the parent ring). (Contributed by Mario Carneiro, 3-Jul-2015.)
 |-  S  =  ( I mPoly  R )   &    |-  H  =  ( Rs  T )   &    |-  U  =  ( I mPoly  H )   &    |-  B  =  ( Base `  U )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  T  e.  (SubRing `  R )
 )   &    |-  W  =  ( I mPwSer  H )   &    |-  C  =  (
 Base `  W )   &    |-  K  =  ( Base `  S )   =>    |-  ( ph  ->  B  =  ( C  i^i  K ) )
 
Theoremressmplbas 16200 A restricted polynomial algebra has the same base set. (Contributed by Mario Carneiro, 3-Jul-2015.)
 |-  S  =  ( I mPoly  R )   &    |-  H  =  ( Rs  T )   &    |-  U  =  ( I mPoly  H )   &    |-  B  =  ( Base `  U )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  T  e.  (SubRing `  R )
 )   &    |-  P  =  ( Ss  B )   =>    |-  ( ph  ->  B  =  ( Base `  P )
 )
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